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The Integers and Division
Division
If a and b are integers with a 0, we say that a
divides b if there is an integer c such that b =a c.
When a divides b we say that a is a factor of b
and that b is a multiple of a. The notation a|b
denotes that a divides b.
Example: Determine whether 3|7 and whether
3|12.
Example: Let n and d be positive integers. How
many positive integers not exceeding n are
divisible by d?
Theorem: Let a, b, c be integers. Then
(i) If a|b and a|c then a|(b+c)
(ii) If a|b then a|bc for all integers c
(iii) If a|b and b|c then a|c
Corollary: If a, b and c are integers such that a|b
and a|c, then a|mb+nc whenever m and n
are integers.
The division algorithm
Theorem: Let a be an integers and d a positive
integer. Then there are unique integers q and r,
with 0=<r<d such that a=dq+r.
Definition: In the equality given in the division
algorithm, d is called the divisor, a is called the
dividend, q is called the quotient, and r is called
the remainder. This notation is used to express
the quotient and remainder:
q=a div d, r=a mod d
Example: What are the quotient and remainder
when 101 is divided by 11?
What are the quotient and remainder when -11
is divided by 3?
Modular Arithmetic
Definition: If a and b are integers and m is a
positive integer, then a is congruent to b
modulo m if m divides a-b. We use the
notation a ´ b (mod m) to indicate that a is
congruent to b modulo m.
Theorem: Let a and b be integers, and let m be a
positive integer. The a ´ b (mod m) if and
only if a mod m = b mod m.
Example: Determine whether 17 is congruent to
5 modulo 6 and whether 24 and 14 are
congruent modulo 6.
Theorem: Let m be a positive integer. The
integers a and b are congruent modulo m if
and only if there is an integer k such that
a=b+km.
Theorem: Let m be a positive integer. If a ´ b
(mod m) and c ´ d (mod m), then a+c ´ b+d
(mod m) and ac ´ bd (mod m)
Corollary: Let m be a positive integer and let a
and b be integers. Then
(a+b) mod m = ((a mod m)+(b mod m)) mod m
And
Ab mod m = ((a mod m)(b mod m)) mod m.
Applications of Congruences
Hashing Functions: The central computer at an
insurance company maintains records for each
of its customers. How can memory locations
be assigned so that customer records can be
retrieved quickly? The solution to this problem
is to use a suitably chosen hashing function.
Records are identified using a key, which
uniquely identifies each customer’s record.
For instance Social Security number.
In practice, many different hashing functions are
used. One of the most common is the function
h(k) = k mod m
Where m is the number of available memory
locations.
Hash functions should be easily evaluated so
that files can be quickly located.
H(064212848) = 064212848 mod 111 = 14
H(037149212)= 037149212 mod 111 = 65
H(107405723)= 107405723 mod 111 = 14
But this location has been already occupied.
However memory location 15, the first
location following memory location 14 is free

Pseudorandom Number
Randomly chosen numbers are often needed for
computer simulations. Different methods have
been devised for generating numbers that have
properties of randomly chosen numbers. Because
numbers generated by systematic methods are
not truly random they are called pseudorandom
numbers.
The most commonly used procedure for generating
pseudorandom numbers is the linear
congruential method. Xn+1 = (a xn +c) mod m